Beilinson–Bernstein localization

In mathematics, especially in representation theory and algebraic geometry, the Beilinson–Bernstein localization theorem relates D-modules on flag varieties G/B to representations of the Lie algebra ${\displaystyle {\mathfrak {g}}}$ attached to a reductive group G. It was introduced by Beilinson & Bernstein (1981).

Extensions of this theorem include the case of partial flag varieties G/P, where P is a parabolic subgroup in Holland & Polo (1996) and a theorem relating D-modules on the affine Grassmannian to representations of the Kac–Moody algebra ${\displaystyle {\widehat {\mathfrak {g}}}}$ in Frenkel & Gaitsgory (2009).